Experiments with cellular structures

本提案的对象是表示理论，结理论和数学物理中感兴趣的五类代数：-经典舒尔代数（与代数群GLn相关），-对称群的群代数，-布劳尔代数，-分拆代数，-最近发现的布劳尔代数的舒尔代数。我们将重点关注其细胞结构的关键和计算可访问的部分，即细胞子商内的乘法，以及关于简单模块，半简单性和从这种乘法中导出的其他基本性质的信息。该项目包括三个主要部分：首先，我们将开发算法和计算机程序来计算相关的结构常数，即使是大型示例。其次，我们将进行实验，特别是通过改变代数的一些离散参数和连续参数。第三，我们将使用实验的输出来制定关于参数的结构和独立性的精确猜测，并为一般结果的证明提供起点。

The objects of this proposal are five classes of algebras of interest in representation theory, knot theory and mathematical physics:– Classical Schur algebras (associated with the algebraic group GLn),– group algebras of symmetric groups,– Brauer algebras,– partition algebras,– the recently discovered Schur algebras of Brauer algebras.We will focus on a crucial and computationally accessible part of their cellular structure, the multiplication inside cellular subquotients, and on the information about simple modules, semisimplicity and other fundamental properties to be derived from this multiplication. The project consists of three main parts: First, we will develop algorithms and computer programs to calculate the relevant structure constants, even for large examples. Secondly, we will run experiments, especially by varying some of the discrete and of the continuous parameters of the algebras. Thirdly, we will use the output of the experiments to formulate precise conjectures about structure and independence of parameters and to get starting points for proofs of general results.